Example: I enter the Asteroid System from the left side (via the green spaceport space) and inspect the First Contact Card per the rules. If it is a culture I want to activate, I now need to get to one of the Open Neutral Spaceports in the Asteroid System to flip and take the first contact card, move the culture token, race goods tokens, factory goods and deed token to the market of the system, and then place two spaceport deed tokens on the green spaceport spaces in the system at the left and rightmost edges?

You have it right about observing the culture from the Green Orbit spaces. And you do have to actually land at one of the two Asteroid System spaceports to actually discover the Culture that is there and collect the IOU, place the Culture marker, Goods and Factory Deed.

But after that, you're off a bit.

First, the Asteroid System Spaceports are NOT Open Spaceports. So, the first turn you land on one, you are limited to the usual one Buy action and one Sell action. If you stay on subsequent turns, then you can do unlimited Buys and Sells. The only Open Spaceports in the game, are the ones you discover through Encounter spaces.

Secondly, the only way to get spaceports on the Green Orbit spaces is to buy a Spaceport Deed from either the Asteroid System Spaceport or an already existing Spaceport in that system. So, with the first Spaceport in the Asteroid System, you usually have to land at one of the Asteroid Systems two Spaceports, buy a Spaceport Deed, THEN you place one of your spaceport markers on any open Green Orbit space. Then if you want to actually use it, you have to fly there on a later turn.